Q-3 The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form,pq what can you say about the prime factor of q?

43.123456789

0.120120012000120000…

43.123456789¯

Solution:

Types of Decimal Expansions

Types of Decimal Expansions

43.123456789

43.123456789 Is terminating

So, it would be a rational number

43.123456789=431234567891000000000

=43123456789(10)9

=43123456789(2×5)9

=4312345678929×59

Hence 43.123456789 is now in the form of pq

And the prime factors of q are in terms of 2 and 5

0.120120012000120000…

0.120120012000120000…Is non terminating and non-repeating

So, it is not a rational number

Therefore, the given number is an irrational number

43.123456789¯

Here 43.123456789¯ is non terminating but repeating decimal

So, it would be a rational number of the form pq and q is not of the form 2n×5m

That is the prime factors of q will also have a factor other than 2 or 5.